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High-fidelity Characterization on Anisotropic Thermal Conductivity of Carbon Nanotube Sheets and on Their Effects of Thermal Enhancement of Nanocomposites
Xiao Zhanga (), Wei Tanb, Fiona Smaila, Michael De Volderc, Norman Fleckb, Adam Boiesa()
a Division of Energy, Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK b Division of Mechanics, Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK c Institute for Manufacturing, Department of Engineering, University of Cambridge, Cambridge, CB3 0FS, UK
ABSTRACT Some assemblies of nanomaterials, like carbon nanotube (CNT) sheet or film, always show outstanding and anisotropic
thermal properties. However, there is still a lack of comprehensive thermal conductivity (𝜅) characterizations on CNT sheets,
as well as lack of estimations of their true contributions on thermal enhancement of polymer composites when used as
additives. Always, these characterizations were hindered by the low heat capacity, anisotropic thermal properties or low
electrical conductivity of assemblies and their nanocomposites. And the transient 𝜅 measurement and calculations were
also hampered by accurate determination of parameters, like specific heat capacity, density and cross-section, which could
be difficult and controversial for nanomaterials, like CNT sheets. Here, to measure anisotropic 𝜅 of CNT sheets directly
with high fidelity, we modified the conventional steady-state method by measuring under vacuum and by infrared camera,
and then comparing temperature profiles on both reference standard material and a CNT sheet sample. The highly anisotropic
thermal conductivities of CNT sheets were characterized comprehensively, with 𝜅/𝜌 in alignment direction as ~95
mW·m2/(K·kg). Furthermore, by comparing the measured thermal properties of different CNT-epoxy resin composites, the
heat conduction pathway created by the CNT hierarchical network was demonstrated to remain intact after the in-situ
polymerization and curing process. The reliable and direct 𝜅 measurement rituals used here, dedicated to nanomaterials,
will be also essential to assist in assemblies’ application to heat dissipation and composite thermal enhancement.
KEYWORDS
Thermal conductivity, Carbon nanotube, Steady state, Composite, Nanomaterial, Heat dissipation
————————————
Corresponding author, Xiao Zhang email: [email protected] ; Adam Boies: [email protected]
2
1. Introduction
Due to the outstanding thermal properties of many
nanomaterial, their corresponding assemblies, such as the
carbon nanotube (CNT) sheet, fibres [11, 24], graphene and
BN foams [4, 19], have been research hotspots for years in
the field of heat dissipation and thermal conduction
enhancement in polymer composites. However, reported
experimental results have repeatedly shown the degradation
of thermal properties when nanomaterials were used to
form assemblies. For instance, thin films made from high
quality CNTs show the in-plane thermal conductivity (𝜅) to
be ~110 W/(K·m) and an improved 770 W/(K·m) for fibres
after stretching and solvent densifying [7]. However, this 𝜅
value is still much lower than that of an isolated SWCNT in
both theoretical simulations (6600 W/(K·m)) [2] and
various experimental results (2000-4000 W/(K·m) [6, 13,
26]) with different samples and measurement methods).
The mechanisms that underlie the large discrepancy
between bulk properties and individual CNTs have not been
well described in literature, motivating detailed study of the
processing and assembly of nanomaterial and their
composite, to better take advantage of their outstanding
thermal properties.
Among CNT assemblies, CNT sheet, with its outstanding
performance, synthesized by use of the continuous floating
catalyst chemical vapor deposition (FCCVD) method [14]
has already become commercial with an industrial
production¶. With single layers of CNT aerogel emanating
from tube furnace, collected on a roller, and finally
densified by compression, a preferential alignment formed
in plane of CNT sheets, as well as a multilayer structure out
of plane. Consequently, CNT sheet shows highly
anisotropic thermal and electrical properties.
However, to comprehensively study the anisotropic thermal
properties in assemblies like CNT sheets, characterizations
were always hindered by the lack of reliable measurement
techniques, due to their incompetence on anisotropic
thermal properties, as well as by low heat capacities and
mass densities of assemblies.
For instance, with the frequently used laser flash technique
[18], researchers deduced 𝜅 from measured thermal
diffusivity (𝛼) with specific heat capacity (𝑐𝑝) and
density (𝜌). However, the determination of an accurate 𝑐𝑝
and 𝜌 value is challenging for nanomaterials and their
composites. This is reflected in the fact that scientific
publications show a very large spread (>20%) of 𝑐𝑝 values
for CNTs [12], which is further complicated for 𝑐𝑝 of their
resulting composites. Moreover, the transient measurement
errors propagate when calculating 𝜅 , resulting in large
uncertainties for nanomaterials. On the other hand, to
measure 𝜅 of nanomaterials directly, traditional steady-
state methods also require modification. For example,
inevitable thermal contact resistance has always led to
erroneously lower 𝜅 values. And nanomaterials’ small
heat capacities and thickness also leave temperature
measurement susceptible to heat dissipation from
temperature probes (e.g. thermocouples). Moreover, high κ
materials require longer samples to maintain a reliable
temperature gradient, which increases the surface area for
convective and radiative heat loss. These complications
introduce non-steady effects, such as fluctuating heat loss
from free convection, which must be avoided to achieve
consistent results.
Furthermore, to evaluate the fillers or additives’ thermal
enhancement effect in a polymer composite, methods like
the transient 3ω method or steady-state Joule-heating
method [27], are not adaptable, due to complicated
electrical resistance change of composites with temperature.
Here we modified the conventional steady-state κ
measurement method, adapting it for nanomaterial
measurement, and demonstrated the applicability of the
technique by measuring CNT sheet and composites. We
limit convective interference by conducting the
measurements under vacuum. And all the temperature
profiles were measured by an infrared camera to prevent
heat dissipation and temperature disturbance from
thermocouples. By taking both radiation and convection
heat transfer into consideration, we measured κ in plane (x
and y directions) with its confidence limit in various
operating environments by stochastic error propagation
calculation. With another steady state method, we
calculated the κ out of plane (z-direction) and the Biot
number. Moreover, with the modified in plane κ
measurement technique, we measured thermal properties of
different CNT-epoxy resin composites to study the CNT
sheet filler’s effect on thermal enhancement for polymer
composites.
2. Experimental Detail and Theoretical Basis for In
Plane Measurement
2.1 Experimental Detail
As shown in Fig. 1 inset, to avoid the disturbance from free
convection on nanomaterial samples, all the measurements
were taken in a cylindrical aluminium vacuum chamber (∅
350 mm × height 300 mm) linked on the sidewall with a
turbomolecule pump (PFEIFFER TPH 240PC). Optical
access was achieved via a sidewall port, fitted with a 3 mm
thick ZnSe optical window with anti-reflection layers on
both sides. The window transmittance was calibrated to be
approximately 92% at 7-14 µm.
A FLIR T650sc infrared camera (640×480 LWIR
resolution), equipped with 7.5-14 µm spectral band optics
(f/1.0, focal length 24.6mm), as well as the ResearchIR
software interface, was used to measure the temperature
profile along samples. The noise-equivalent temperature
difference (NETD) of <20 mK guaranteed high precision
temperature resolution along the sample axis.
In Fig. 1, copper blocks were used as heat sinks at both hot
and cold ends, whose temperatures were controlled by
metal foil heaters (Kapton insulated) and K-type tiny-
thermocouples (OMEGA SA3-K). The heat sink at the hot
end (HotEnd) was thermally isolated by a 20 mm thick
3
PTFE buffer layer. In contrast, the heat sink at the cold end
(ColdEnd) was thermally linked to the vacuum chamber
with an aluminium adaptor. The metallic reference material
and nanomaterial sample (CNT sheet strip) were inserted
into the HotEnd and ColdEnd, respectively. The other end
of nanomaterial sample was attached onto the reverse side
of the reference material with a thin layer of high thermal
conductivity silver paint, forming the junction section (3 –
5 mm). An XYZ translation stage at the ColdEnd eliminated
the distortion on samples, to avoid disturbance of the
thermal measurement.
For CNT sheet samples, we cut one large CNT sheet
(thickness ~70-100 μm) in different directions to obtain
different anisotropic CNT sheet strip samples, with the
length ~50 mm in the x(y)-direction, and the width of 5 mm
in the y(x)-direction.
By properly selecting a reference material with suitable
thermal conductivity and geometry, comparable
temperature gradients were achieved on both the reference
material and the sample. For metallic materials, the
emissivity was modified with a thin layer of black paint to
guarantee a reliable temperature reading.
The measurement procedure began when all components
reached a stable temperature (with temperature change no
more than 0.1 K over 10 min) under vacuum. Then, both
the temperature profile along the reference material and
nanomaterial were recorded by infrared camera with the
mean temperature taken along the sample width (y
direction). Temperatures of the ColdEnd, HotEnd, chamber
wall and the atmosphere were simultaneously recorded by
tiny thermocouples.
2.2 Theory Basis
Based on the low Biot number (𝐵𝑖 ≪ 1 ) calculated in
section 4, we used the one-dimensional heat conduction
equation (Eq. 1), accounting for convection and radiation
heat loss, to model the temperature profile of the sample,
𝜅𝐴𝑑2𝑇
𝑑𝑥2− ℎ𝑐𝑃(𝑇 − 𝑇𝑔) − 𝜀𝜎𝑃(𝑇4 − 𝑇∞
4) = 0 (1)
𝜅 represents the thermal conductivity along the
temperature gradient (x direction), 𝐴 represents the
sample’s cross-section area orthogonal to the heat transfer
(y-z plane), ℎ𝑐 is the convective heat transfer coefficient,
𝑃 is the sample’s surface isothermal perimeter from which
the convection (radiation) heat was transferred to the
surrounding gas (environment), 𝜀 was the sample’s
emissivity, 𝜎 was the Stefan–Boltzmann constant, and T
represents the temperatures of the system, gas (𝑇𝑔) and
environment (𝑇∞).
In previous reports, to solve Eq. 1, temperature at the two
ends have been frequently used as the boundary conditions
[25]. However, this choice could make 𝜅 sensitive to
temperature errors. Here, as shown in Fig. 2a, we followed
the reported dual-mode method [15] to use the heat flux
𝑄𝑖𝑛 at the HotEnd as the first boundary condition (Eq. 2),
which is proportional to the temperature gradient. The
second boundary condition was the temperature at the
ColdEnd (Eq. 3):
−𝜅𝐴𝑑𝑇
𝑑𝑥|𝑥=0
= 𝑄𝑖𝑛 (2)
𝑇(𝑥 = 𝐿) = 𝑇𝐶𝐸 (3)
where 𝐿 represents the length of the sample. This method
makes the fitting process much more robust against
parameter errors.
By maintaining the temperature ranges of sample and
reference material’s cold tail (in Fig. 2b, plot with orange
background) within 10 K ranges, it was reasonable to
consider 𝜅, ℎ𝑐, 𝜀 as fixed values. Moreover, to simplify
calculation, the radiation term was replaced by its 1st order
Taylor polynomial‡:
𝑇4 − 𝑇∞4 ≈ 4𝑇𝑚𝑒𝑎𝑛
3(𝑇 − 𝑇∞) (4)
Above 𝑇𝑚𝑒𝑎𝑛 was the mean temperature of target sample
with surrounding shield. Within 10 K above room
temperature (~297 K), it is reasonable to treat 𝑇𝑚𝑒𝑎𝑛 as a
constant when solving the Eq. 1.
The solution for Eq. 1 is as follows:
𝑇 =(𝑇𝐶𝐸−
𝛽
𝛼) 𝑐𝑜𝑠ℎ(𝜔𝑥)+
𝑄𝑖𝑛𝜔𝜅𝐴
𝑠𝑖𝑛ℎ(𝜔(𝐿−𝑥))
𝑐𝑜𝑠ℎ(𝜔𝐿)+
𝛽
𝛼 (5)
where 𝛼 = ℎ𝑐𝑃 + 𝐻𝑟𝜀𝑃 , 𝛽 = ℎ𝑐𝑃𝑇𝑔 +𝐻𝑟𝜀𝑃𝑇∞ , 𝜔 =
√𝛼
𝜅𝐴, 𝐻𝑟 = 4𝑇𝑚𝑒𝑎𝑛
3𝜎
Based on Eq. 5, 𝜅 could be calculated by fitting the
experimental temperature profile of the sample with all
experimental parameters.
Considering energy conservation at the junction section,
𝑄𝑖𝑛 could be determined by deducting radiation (𝑄𝑟𝑎𝑑) and
Figure 1 Schematic illustration of thermal conductivity
measurement stage with copper hot and cold end, brass strip
reference material and carbon nanotube sheet strip sample. Insert
shows infrared camera measuring temperature through a sidewall
port on the aluminium vacuum chamber (∅ 350 mm × height 300
mm).
4
convection heat loss (𝑄𝑐𝑜𝑛𝑣) from the heat flow 𝑄𝑡𝑜𝑡 into
the junction section (out of the reference sample).
𝑄𝑖𝑛 = 𝑄𝑡𝑜𝑡 − 𝑄𝑐𝑜𝑛𝑣 − 𝑄𝑟𝑎𝑑 (6)
2.2.1 Stochastic error propagation calculation and
sensitivity analysis
During the fitting process, all geometric, thermal, and
optical parameters had inherent uncertainties, including
experimental random errors and systematic errors. All these
errors led to the deviation of calculated κ from the true κ
value.
Considering the practical engineering application demand,
referential errors were also calculated by the stochastic
error propagation method, generating confidence limits for
each calculated κ value, as shown in Fig. 2e. Briefly,
during every round of iteration, 𝜅 was fitted with a new set
of random parameters which were generated based on
parameters’ expected values, errors, bounds and
distribution style.
Meanwhile, by using Spearman's rank correlation
coefficient, the correlation analysis between 𝜅 and
parameters’ uncertainties was also conducted. Thus, the
thickness and 𝜅 of the reference sample, as well as ℎ𝑐 were determined to reduce systematic errors during
experiments.
2.2.2 Thermal conductivity normalized by volume
density
For porous nanomaterials like the CNT sheet, it is
challenging to obtain true thickness or solid volume.
Incorrect thickness or volume is one of the major sources
of reported κ values variation.
To obtain more accurate results, we adopted thermal
conductivity normalized by solid volume density (κ/ρ), as
the primary dimensions of reported results [W∙m2/(K∙kg)].
𝜅/𝜌 was deduced from the measured value κA in Eq. 5 as
follows: 𝜅
𝜌= 𝜅 ÷
𝑚
𝑙∙𝐴′≈ 𝜅𝐴 ÷
𝑚
𝑙 (7)
Here, 𝑚/𝑙 was the mass per meter, and 𝐴′ was the solid
cross-sectional area in CNT sheet excluding all voids.
To compare our results with other literature values, thermal
conductivity 𝜅 could be deduced by incorporating a
sample solid density. Regarding the CNT sheet, the true
solid density of porous sheet 𝜌 is around ~1.5 g/cm3 [1].
As can be seen from Eq. 5 and 7, 𝜅𝐴 is the value we
measured by experiments and calculated without
assumption of any material parameter. Here we make a
reliable assumption: 𝐴′ ≈ 𝐴 , i.e., the cross-sectional area
used in the density calculation is also the area through
which heat is conducted. By this method, only reliable
values conforming to the experiment have been used in the
calculation of 𝜅. The measured 𝜅 value describes the heat
conduction solely in the solid CNT accounting for the
nanomaterials’ porous structure.
2.3 Calibration of emissivity and transparency
All samples tested did not exhibit pure blackbody
emissivities (𝜀 < 1) . By using an infrared camera, the
accuracy of the detected temperature primarily relies on the
accurate measurement of a samples’ emissivity, as well as
the window’s transmittance at the measured temperature.
𝜀 of all samples have been determined by following the
Figure 2 (a) Infrared temperature contour image during experiment, brass and copper strips were used as reference and target sample
respectively, with a junction of approximately 5 mm between them; (b) Heat flow (𝑄𝑡𝑜𝑡) into junction was calculated by linear fitting,
based on the low temperature tail of the reference sample (plot with orange background); (c) thermal conductivity κ was deduced by
fitting of the temperature profile along the target sample; (d) the emissivities of various samples were adjusted to get accurate temperature
reading; (e) with Stochastic error propagation modelling, κ/ρ of copper was deduced to be 45±4 mW∙m2/(K∙kg).
5
procedures shown in ASTM E1933-14. Briefly, we attached
various samples onto a large copper heat sink with highly
thermal conductivity silver paint. The temperature of the
heat sink was more than 20 K higher than that of the
surrounding environment. The emissivity could be
determined when the detected temperature of sample
equaled that of the heat sink upon adjustment of 𝜀. As can
be seen from Fig. 2d, due to material difference and surface
roughness, the colors in the infrared image were different.
The calibrated 𝜀 of CNT samples was approximately
0.7±0.1.
With ε of the samples and temperature of the chamber, the
transmittance of window was determined by following
procedures shown in ASTM E1897-14, and was found to be
0.92 at 7-14 µm at room temperature.
2.4 Calibration of convection heat transfer coefficient
In high vacuum environments, heat dissipation mainly
comes from radiation which can be evaluated from the
Stefan–Boltzmann equation. To improve 𝜅 measurement
precision and compare the heat loss constituents from
radiation and convection, we also measured the convective
heat transfer coefficient (still represented by ℎ𝑐) under 3
different gas pressures (6 μbar with turbomolecule pump,
0.2 mbar with only mechanical pump and atmospheric
pressure), as shown in Table 1. With ℎ𝑐 calibrated on the
standard material with known thermal conductivity under
different pressures, 𝜅 of the standard sample and
nanomaterials were then calculated.
It is common knowledge that ℎ𝑐 varies with gas density,
viscosity, thermal conductivity, specific heat, and flow
conditions [3]. To calibrate ℎ𝑐 , a brass strip (0.025 mm
13505 Brass foil, Alloy 260 AlfaAesar) was mounted
directly between the HotEnd and ColdEnd. The
temperature profile along the whole brass strip was
measured. The low temperature region (within 10 K) was
chosen as the “sample” section, and a short adjacent region
with higher temperature was used as the “reference material”
to calculate 𝑄𝑖𝑛. There was no junction section between the
“reference material” and the “sample” sections. By
repeatedly pumping and venting with different gases, and
based on the reported κ of brass alloy 260 of 110 W/(K·m)
[3], the ℎ𝑐 of air, He and Ar under different pressures have
been calibrated as shown in Table 1.
Under atmospheric pressure, a thermal boundary layer will
develop in gas adjacent to the vertical heated sample. Based
on the calculated Rayleigh number (Ra, Method A-1 in the
ESM), the Ra was around 30-400, which was far below the
critical Ra=109 to develop a turbulent boundary layer. Thus,
a stable laminate boundary layer will develop along
samples. Based on empirical functions [3, 5], average
Nusselt number (𝑁𝑢̅̅ ̅̅ ) and average convective coefficient
(ℎ𝑐̅̅ ̅) could be deduced (Method A-1 in the ESM). Besides,
if treated as a constant, ℎ𝑐 at 1 bar was calibrated to be
~8.6 W/(K·m2) for air, close to the value calculated by
empirical correlations, as shown in Table 1. In contrast, the
Table 1 Convection heat transfer coefficient calibrated by
standard brass material
Gas
ℎ𝑐
at 6 μbar
[W/(K·m2)]
ℎ𝑐
at 0.2 mbar
[W/(K·m2)]
ℎ𝑐
at 1 bara
[W/(K·m2)]
Empirical
ℎ𝑐̅̅ ̅ at 1 bar
[W/(K·m2)]
Air 0.068±0.140 0.75±0.55 8.6±2.2 7.1-10.8
He 0.059±0.131 1.07±0.63 16.4±2.9 9.1-16.9
Ar 0.057±0.141 0.50±0.46 4.5±1.4 2.9-5.8 a The free convection coefficient ℎ𝑐 at 1 bar should change with
sample’s width and surface temperature. Herein, all data was
calibrated based on a brass strip with 7 mm in width within 10 K
above room temperature (~297 K)
higher ℎ𝑐 of helium gas resulted in the target sample
decreasing to the gas temperature over a shorter length
range. This was a consequence of the different gas densities
at 1 bar and a much different 𝜅 at 300 K (Table S1. in the
ESM).
Under 6 μbar, ℎ𝑐 of argon and helium showed no obvious
difference to air. This may be the result of the molecular
ballistic motion becoming increasingly dominant under
vacuum. Under 0.2 mbar, ℎ𝑐 of air was 0.75±0.55
W/(K∙m2), which was similar to the reported 0.8 W/(K·m2)
at 0.18 mbar [21]. The mean free path ( 𝜆𝑚𝑓𝑝 ) of air
molecules is ~0.34 mm, which is large enough to avoid air
molecules being constrained in micro/nanoscale surface
geometry [3]. Similar ℎ𝑐 relationship changing with gases
with that under atmospheric pressure, manifests the
transition from viscous flow to molecule flow.
Under atmospheric pressure, the mean free path (𝜆𝑚𝑓𝑝) for
air, is ~66 nm. Considering the influence on ℎ𝑐 by various
surface geometry of samples, as well as multiple choices of
empirical correlations, although 𝜅 could also be roughly
measured under atmospheric pressure with empirical ℎ𝑐 , most of our experiments have been finished under vacuum
condition around 6 μbar, under which 𝜆𝑚𝑓𝑝 ~11mm. For
nanomaterials with high emissivity (like CNT sheets with
𝜀~0.7), the typical radiation heat loss coefficient (𝐻𝑟𝜀) was
approximately 5±1 W/(K∙m2). Since ℎ𝑐 is less than 3% of
𝐻𝑟𝜀 , the major heat dissipation route under 6 μbar was
indeed radiation. Thanks to less heat loss from convection,
temperature profile could be measured more precisely.
Moreover, much smaller ℎ𝑐 under vacuum giving a
smaller Bi number, also guarantees a more uniform
temperature distribution on sample cross-section (section 4).
3. Results and Discussion for In Plane Measurement
3.1 Setup verification using standard materials
To verify the feasibility of our setup, as shown in Fig. 2a, a
brass strip (0.13±0.01 mm thick, 13504 Alloy 260 Brass foil,
AlfaAesar) and a copper strip (0.025±0.003 mm thick,
46986 99.8% Copper foil, AlfaAesar) were used as the
reference material and the sample, respectively.
Based on reported 𝜅=110 W/(K∙m) of brass alloy 260 [3],
the 𝜅/𝜌 of copper was measured under vacuum as
45.5±3.7 mW·m2/(K∙kg), i.e. 𝜅 =407.5±33.7 W/(K·m)
6
(with density of 8.96 g/cm3), compared to the reported 401
W/(K·m) [3]. The main error came from the error of
standard material’s thickness (0.025±0.003 mm). However,
when measured at atmospheric pressure, and using the
empirical ℎ𝑐 of free convection, the 𝜅/𝜌 of copper was
measured as 43.6±4.8 mW∙m2/(K∙kg), i.e. 𝜅 =390.5±43.0
W/(K·m). Due to the reasons mentioned in the end of
section 2.4, the following measurements were all conducted
under 6 μbar vacuum condition to obtain more consistent
results.
3.2 In Plane Thermal Conductivity of Thick CNT Sheets
All of the CNT sheet samples tested in our system were
synthesized by use of the continuous FCCVD method [14],
supplied by Tortech Nanofibers Ltd¶. Single layers of CNT
aerogel emanating from tube furnace were collected on a
roller, and finally densified by compression. Due to the gas
flow drag force during the synthesis process and a small
tension force applied during the collecting process, CNT
bundles formed a preferential alignment direction in CNT
sheets. Here the directions parallel with and perpendicular
to the alignment direction in the CNT sheet are called the
x-direction and y-direction, respectively. By cutting in
different directions from one large CNT sheet, we obtained
CNT strip samples with the length in the x(y)-direction, and
the width in the y(x)-direction. The sample was linked with
0.025 mm thick reference brass foil strips with junction
section approximately 5 mm in length direction. Based on
the above steady-state measurement routine, the results
showed that, in the x-direction, the as-received CNT sheet
reached 𝜅/𝜌 =94.9±20.1 mW∙m2/(K∙kg) due to simple
densification via compression. In the y-direction, 𝜅/𝜌 =84.4±17.5 mW∙m2/(K∙kg). The lower value in the y-
direction was due to fewer CNTs bundles in the heat
conduction pathway compared to the x-direction.
Compared to the result in section 3.1, the 𝜅/𝜌 of CNT
sheet was approximately 2 times of that of pure copper. This
high value also confirmed our motivation to develop a
steady-state method for highly thermal conductivity
materials.
Based on the solid density of SWCNT 1.5 g/cm3 [1], 𝜅
could be deduced to be 142±30 W/(K·m) in the x-direction
and 𝜅=127±26 W/(K·m) in the y-direction, both of which
are still much lower than the value of isolated SWCNT
(𝜅=~3000 W/(K·m) [25]). The difference originates mainly
from the disorientation of CNT in the sheet (as shown in
Fig. S-1 in the Electronic Supplementary Material (ESM))
and the phonon quenching at boundaries, between
impurities, as well as defects etc. By using the material
density of solid CNT, the above conductivity reported here
purposely omits the air voids within the CNT material both
within an individual CNT and between CNTs.
The CNT sheet can be regarded as a composite of CNT and
air inside. In the experiment, 𝜅𝐴 was measured. The
effective thermal conductivity 𝜅𝐶𝑁𝑇𝑒𝑓𝑓
could be calculated by
estimating a rectangular cross-sectional area, that is, to treat
the sheet as a homogenous material. The relationship
between 𝜅𝐶𝑁𝑇𝑒𝑓𝑓
and the true thermal conductivity of CNT
network could be roughly described as follows [9],
𝜅𝐶𝑁𝑇𝑒𝑓𝑓
= 𝜙𝜅𝐶𝑁𝑇 + (1 − 𝜙)𝜅𝑎𝑖𝑟 (8)
Although the volume fraction 𝜙 of air in the CNT sheet
was higher than that of CNT (as shown in Table 2), since
𝜅𝑎𝑖𝑟 is too low in comparison to 𝜅𝐶𝑁𝑇, the contribution of
air in the thermal dissipation is negligible. Here by using
the calculation method shown in section 2.2.2, all the κ
results reported here were 𝜅𝐶𝑁𝑇 instead of 𝜅𝐶𝑁𝑇𝑒𝑓𝑓
.
Table 2. Thermal conductivity in x-direction of composites with different epoxy content.
Samples 𝑡𝐶𝑁𝑇
[μm]
𝑡𝑒𝑓𝑓
[μm]
𝜌𝑒𝑓𝑓
[g/cm3]
CNT
wt%
CNT
vol%
Epoxy
vol%
Air
vol%
𝜅𝐴/𝑤/𝑡𝐶𝑁𝑇
[W/(K·m)] 𝜅𝑐𝑝𝑒𝑓𝑓𝑎
[W/(K·m)]
𝜅𝐶𝑁𝑇𝑏
[W/(K·m)]
Pure CNT 111 113 0.33 100% 19.5% 0% 80.5% 33.1±6.4 32.6±6.3 167±32
Composite #1 107 89 0.44 98.9% 25.2% 0.4% 74.4% 32.4±6.1 39.0±7.4 155±29
Composite #2 106 86 0.47 92.8% 25.5% 3.0% 71.5% 33.5±6.1 41.4±7.6 162±30
Composite #3 96 73 0.72 65.2% 25.6% 21.0% 53.4% 34.6±6.6 45.2±8.6 176±33
Composite #4 98 62 1.21 44.0% 30.3% 59.1% 10.6% 32.9±6.8 51.6±10.7 170±35
Composite #5 124 129 1.23 21.4% 14.7% 83.2% 2.1% 29.6±5.7 28.4±5.5 191±35
Composite #6 131 149 1.19 19.8% 13.7% 85.5% 0.8% 26.6± 5.1 23.5±4.5 169±31
a 𝜅𝑐𝑝𝑒𝑓𝑓
of composites were calculated by dividing experimental results 𝜅𝐴 with sample strips’ width w and 𝑡𝑒𝑓𝑓 of composites measured by a micrometer;
b κCNT is calculated by solving 𝜅𝑐𝑝𝑒𝑓𝑓
= 𝜙𝐶𝑁𝑇𝜅𝐶𝑁𝑇 + 𝜙𝑒𝑝𝑜𝑥𝑦𝜅𝑒𝑝𝑜𝑥𝑦 + 𝜙𝑎𝑖𝑟𝜅𝑎𝑖𝑟 , here ϕ is the volume fraction and 𝜅𝑒𝑝𝑜𝑥𝑦 =0.35 W/(K·m), 𝜅𝑎𝑖𝑟 =0.026
W/(K∙m) For pure CNT, 𝜅𝐶𝑁𝑇 could also be calculated by Eq.7. which was 153±31 W/(K·m).
7
3.3 Thermal Conductivity of CNT thick sheet
reinforced polymer composite
The CNT sheet reinforced composite was fabricated by
submerging CNT sheets into epoxy/acetone mixtures
(varying the epoxy concentration) for 1 minute to make a
CNT sheets prepreg. These resin-impregnated CNT
sheets were then cured for 3 h at 120 °C in an oven under
pressure of 6 bar, followed by oven cooling to room
temperature (Figure S-2 in the ESM).
Here with the same thermal conductivity measurement
technique, we measured the 𝜅𝐴 of cured CNT
sheet/epoxy composite with different epoxy content. By
using a brass foil reference material, the apparent
temperature gradient could be recorded along the
composite length.
As heterogeneous materials, composites’ density increase
roughly with the increasing of epoxy content.
Consequently, the routine of deducing 𝜅 from 𝜅/𝜌
mentioned in section 2.2.2 may also include additional
errors. To illustrate the influence of epoxy on the heat
conduction ability of the composite, we can compare the
value of 𝜅𝐴/𝑤/𝑡𝐶𝑁𝑇 , that is 𝜅𝐴 normalized by CNT
strips’ width (w ) and thickness of CNT sheet (𝑡𝐶𝑁𝑇 ,
thickness of CNT sheet measured by micrometer before
submerged by epoxy).
As shown in Table 2 and Fig S-3a, 𝜅𝐴/𝑤/𝑡𝐶𝑁𝑇 of the
composite remained nearly unchanged from adding small
amount of epoxy resin on CNT sheet to totally
submerging CNT in epoxy. Considering epoxy’s low
thermal conductivity, the preservation of 𝜅𝐴/𝑤/𝑡𝐶𝑁𝑇
from pure CNTs to composites indicated that polymer
encapsulation of the CNTs has not degraded heat
conduction of the composites. Increased phonon
scattering at the polymer matrix and CNTs [17] were not
detectable here. As shown in Table 2, the 𝜅𝐶𝑁𝑇 deduced
from thermal conductivity of composites 𝜅𝑐𝑝𝑒𝑓𝑓
also
indicates a similar trend with 𝜅𝐴/𝑤/𝑡𝐶𝑁𝑇. These results
indicate that the heat conduction pathway in the
composite had not been degraded after the in-situ
polymerization and curing process [20].
Furthermore, 𝜅𝑐𝑝𝑒𝑓𝑓
of composites increased
monotonously with volume fraction of CNT sheet
increase in composites (Fig S-3b and Table 2.). With
maximum CNT volume fraction as ~30%, the thermal
conductivity reached a much high value relative to typical
CNT/epoxy composites, 𝜅𝑐𝑝𝑒𝑓𝑓
=51.6±10.7 W/(K·m) as
compared to 0.24-5.5 W/(K·m) [10]. The high thermal
conductivities of the CNT sheet/epoxy composite result
from the interconnected CNT bundles hierarchical
network produced from FCCVD which offers a
continuous heat conduction pathway (Figure S-1). In the
sheet, the interface between CNTs is longitudinal,
providing 1D linear contact, which reduces the thermal
interface resistance between CNTs when compared to the
point contact found in short CNTs reinforced composites
[10]. The geometry of the CNT sheet composite is similar
to that of the synergistic thermal enhancement effect of
the CNTs and graphene nanoplates [23]. By contrast, in
CNTs powder reinforced composites, CNTs transported
heat through overcoming CNT-Polymer thermal interface
resistance [8]. Additionally, by compressing the CNT
aerogel and in-situ polymerization, the CNT volume
fraction in our composites (>20%) is much higher than
that of CNT powder reinforced composites [10, 16], and
therefore delivers high thermal conductivity per unit
volume.
4 Out of Plane Thermal Conductivity Measurement
and Biot Number Calculation
As mentioned in section 3.2, to synthesize the CNT sheet
by using the continuous FCCVD method, the CNT
aerogel was collected layer upon layer onto a roller, and
finally densified by compression. Consequently, in the out
of plane direction (z-direction) of the CNT sheet, there are
many interfaces between CNT aerogel layers. Hence, 𝜅
in the z-direction was envisaged much lower than that of
the x/y-direction.
As a result, it was necessary to confirm the assumptions
of theory mentioned in section 2.2, i.e. one-dimensional
heat conduction and uniform temperature distribution
within nanomaterials assembly. Here we calculated the Bi
number, which provides a criterion of the temperature
drop in the solid with depth, relative to the temperature
difference between the solid’s surface and the
environment, under convective and radiative transfer [3].
Figure 3 (a) Schematic depicting the temperature profiles of Bi-
substrate technique [22]; (b) Infrared temperature contour image
during measurement by using the modified steady-state Bi-
substrate technique to measure the z-direction κ of pure CNT
sheets; (c, d) Linear fitting and stochastic error propagation
calculation of data from 3 samples with different thickness,
was measured as 108±4 mW/m/K.
8
In particular, if Bi≪0.1, the resistance to conduction
within the solid is much less than the resistance to heat
loss to the surroundings.
According to Fig. 2(a), Bi along the width and z-direction
should have forms as follows [3]:
𝐵𝑖𝑤 = 𝑤(𝑤 + 𝑡)(ℎ𝑐 +𝐻𝑟𝜀)/(2𝑡𝜅𝑦) (9)
𝐵𝑖𝑧 = 𝑡(𝑤 + 𝑡)(ℎ𝑐 + 𝐻𝑟𝜀)/(2𝑤𝜅𝑧) (10)
Above, 𝑤 and 𝑡 are the width and thickness of the
sample, 𝜅𝑤 and 𝜅𝑧 were the thermal conductivities in
the width-direction and z-direction. In section 3.2, we
have already measured thermal conductivity in the x/y-
direction. To measure the low thermal conductivity in the
z-direction, we use another steady-state technique, the Bi-
substrate technique [22]. Briefly, as shown in Fig. 3(a), 2
substrates were used as heat flux meters, 𝜅𝑧 and
interface thermal resistance could be calculated by linear
fitting data obtained from samples with different
thickness by using the following equation: ∆𝑇
𝑞=
∆𝑧
𝜅𝑧+
2
𝑅𝑖 (11)
where ∆𝑇 was the total temperature drop, 𝑞 was heat
flux through sample, ∆𝑧 was samples’ thickness, and 𝑅𝑖 was the interface thermal resistance.
As shown in Fig. 3(b), temperature profiles along copper
heat flux meters above and beneath thin CNTs samples
were recorded by infrared camera. In Fig. 3(c, d), by
linear fitting data from CNT sheets with thicknesses of 78,
85, 92, 147, 232, 305 and 374 μm, and stochastic error
propagation modelling, we measured 𝜅𝑧 of CNT sheets
as 108±4 mW/(K·m) with 𝑅𝑖 ~4990 W/(K∙m2).
Combined with 𝜅𝑥 and 𝜅𝑦 measured in section 3.2, we
calculated Bi numbers under vacuum and atmospheric
pressure.
Under 6 μbar,𝐵𝑖𝑤<7.2×10-3≪0.1 and𝐵𝑖𝑧<1.7×10-3≪0.1.
This small Bi verified the assumption of both 1D heat
conduction and uniform temperature on cross-section. In
Fig. 2a, the small value of 𝐵𝑖𝑤 could also be confirmed
by the uniform temperature across the width of both
samples.
Under atmospheric pressure, the 𝐵𝑖𝑤<1.9×10-2<0.1 and
𝐵𝑖𝑧 <4.5×10-3≪0.1. These results suggested that even
under atmospheric pressure, our modified steady-state
method could also be used. And smaller Bi numbers under
vacuum guaranteed a more uniform temperature
distribution in materials, especially for thin materials with
a poor out of plane thermal conductivity or anisotropic
thermal conductivity.
Conclusions
With modified steady-state technique and calibrated
radiation and convection heat transfer coefficients with
standard material, we directly measured the in plane
thermal conductivity of CNT sheets. With another steady
state method, we calculated the thermal conductivity out
of plane. Results illustrated the anisotropic thermal
conductivity, with in alignment direction, perpendicular
to alignment direction and out of plane direction values as
142.3±30.1 W/(K·m), 126.6±26.2 W/(K·m) and 108±4
mW/(K·m), respectively.
Furthermore, the thermal conductivity of the CNT sheet
reinforced polymer composites could also be measured
using the same in plane method, as 51.6±10.7 W/(K·m)
when the volume fraction of CNT sheet reached ~30%.
Thanks to the compatibility of the modified technique to
both insulating and highly electrically conductive
samples, we showed that the CNT matrix retained its heat
conduction pathway in polymer composites.
Our measurements rituals can be adapted to other highly
conductive nanomaterials assemblies and their
composites, especially materials with small heat
capacities, or anisotropic properties, or low electrical
conductivity. Thus, it will become an important
complement to existing commercial κ measurement.
Acknowledgements This work was supported by EPSRC project ‘Advanced
Nanotube Application and Manufacturing (ANAM)
Initiative’ [grant numbers EP/M015211/1].
The authors thank Tortech Nano Fibers Ltd for offering
CNT sheet materials. The authors also thank Dr. Sarah
Stevenson, Brian Graves, Dr. Jean de La Verpilliere, Dr.
Christian Hoecker and Joe Stallard for their kind support
and useful discussion, and Ms Nicola Cavaleri for writing
assistance.
Declarations of interest: none.
Supplementary Material: details of convective heat loss
coefficient under atmospheric pressure, SEM images and
thermal conductivities of composites are available in the
online version of this article at
https://doi.org/10.17863/CAM.18098
References ‡ When experiment is operated at room temperature and T ≤60℃, the deviation of this simplification is quite small:
(𝑇4−𝑇∞
4 )−[4𝑇𝑚𝑒𝑎𝑛3 (𝑇−𝑇∞)]
𝑇4−𝑇∞4 < 0.3%
¶ More details about the CNT sheet could be found at
http://tortechnano.com/tortech-nano-fibers
[1] Behabtu N, Young C C, Tsentalovich D E,
Kleinerman O, Wang X, Ma A W K, Bengio E A,
ter Waarbeek R F, de Jong J J, Hoogerwerf R E,
Fairchild S B, Ferguson J B, Maruyama B, Kono
J, Talmon Y, Cohen Y, Otto M J and Pasquali M
2013 Strong, Light, Multifunctional Fibers of
Carbon Nanotubes with Ultrahigh Conductivity
9
Science 339 182-6
[2] Berber S, Kwon Y K and Tomanek D 2000
Unusually high thermal conductivity of carbon
nanotubes Phys. Rev. Lett. 84 4613-6
[3] Bergman T L 2011 Introduction to Heat Transfer:
Wiley)
[4] Chen J, Huang X, Zhu Y and Jiang P 2016
Cellulose Nanofiber Supported 3D
Interconnected BN Nanosheets for Epoxy
Nanocomposites with Ultrahigh Thermal
Management Capability Advanced Functional
Materials 1604754-n/a
[5] Churchill S W and Chu H H S 1975
CORRELATING EQUATIONS FOR
LAMINAR AND TURBULENT FREE
CONVECTION FROM A VERTICAL PLATE
Int. J. Heat Mass Transf. 18 1323-9
[6] Fujii M, Zhang X, Xie H Q, Ago H, Takahashi
K, Ikuta T, Abe H and Shimizu T 2005
Measuring the thermal conductivity of a single
carbon nanotube Phys. Rev. Lett. 95 4
[7] Gspann T S, Juckes S M, Niven J F, Johnson M
B, Elliott J A, White M A and Windle A H 2017
High thermal conductivities of carbon nanotube
films and micro-fibres and their dependence on
morphology Carbon 114 160-8
[8] Hu L, Desai T and Keblinski P 2011 Thermal
transport in graphene-based nanocomposite
Journal of Applied Physics 110 5
[9] Hu X J, Padilla A A, Xu J, Fisher T S and
Goodson K E 2006 3-omega measurements of
vertically oriented carbon nanotubes on silicon J.
Heat Transf.-Trans. ASME 128 1109-13
[10] Ji T X, Feng Y Y, Qin M M and Feng W 2016
Thermal conducting properties of aligned
carbon nanotubes and their polymer composites
Compos. Pt. A-Appl. Sci. Manuf. 91 351-69
[11] Jiang S H, Liu C H and Fan S S 2014 Efficient
Natural-Convective Heat Transfer Properties of
Carbon Nanotube Sheets and Their Roles on the
Thermal Dissipation ACS Appl. Mater.
Interfaces 6 3075-80
[12] Kabo G J, Paulechka E, Blokhin A V, Voitkevich
O V, Liavitskaya T and Kabo A G 2016
Thermodynamic Properties and Similarity of
Stacked-Cup Multiwall Carbon Nanotubes and
Graphite J. Chem. Eng. Data 61 3849-57
[13] Kim P, Shi L, Majumdar A and McEuen P L 2001
Thermal transport measurements of individual
multiwalled nanotubes Phys. Rev. Lett. 87
[14] Li Y L, Kinloch I A and Windle A H 2004 Direct
spinning of carbon nanotube fibers from
chemical vapor deposition synthesis Science 304
276-8
[15] Mahanta N K and Abramson A R 2010 The dual-
mode heat flow meter technique: A versatile
method for characterizing thermal conductivity
Int. J. Heat Mass Transf. 53 5581-6
[16] Marconnett A M, Yamamoto N, Panzer M A,
Wardle B L and Goodson K E 2011 Thermal
Conduction in Aligned Carbon Nanotube-
Polymer Nanocomposites with High Packing
Density ACS Nano 5 4818-25
[17] Mayhew E and Prakash V 2014 Thermal
conductivity of high performance carbon
nanotube yarn-like fibers Journal of Applied
Physics 115 9
[18] Parker W J, Jenkins R J, Abbott G L and Butler
C P 1961 Flash Method of Determining Thermal
Diffusivity, Heat Capacity, and Thermal
conductivity Journal of Applied Physics 32
1679-&
[19] Pop E, Varshney V and Roy A K 2012 Thermal
properties of graphene: Fundamentals and
applications MRS Bulletin 37 1273-81
[20] Raravikar N R, Schadler L S, Vijayaraghavan A,
Zhao Y P, Wei B Q and Ajayan P M 2005
Synthesis and characterization of thickness-
aligned carbon nanotube-polymer composite
films Chem. Mat. 17 974-83
[21] Saidi M and Abardeh R H 2010 Air pressure
dependence of natural-convection heat
transfer(vol 2) p 1-4
[22] Tan J C, Tsipas S A, Golosnoy I O, Curran J A,
10
Paul S and Clyne T W 2006 A steady-state Bi-
substrate technique for measurement of the
thermal conductivity of ceramic coatings Surf.
Coat. Technol. 201 1414-20
[23] Yu A P, Ramesh P, Sun X B, Bekyarova E, Itkis
M E and Haddon R C 2008 Enhanced Thermal
Conductivity in a Hybrid Graphite Nanoplatelet
- Carbon Nanotube Filler for Epoxy Composites
Adv. Mater. 20 4740-+
[24] Zhang G, Jiang S H, Yao W and Liu C H 2016
Enhancement of Natural Convection by Carbon
Nanotube Films Covered Microchannel-Surface
for Passive Electronic Cooling Devices ACS
Appl. Mater. Interfaces 8 31202-11
[25] Zhang X, Song L, Cai L, Tian X Z, Zhang Q, Qi
X Y, Zhou W B, Zhang N, Yang F, Fan Q X,
Wang Y C, Liu H P, Bai X D, Zhou W Y and Xie
S S 2015 Optical visualization and polarized
light absorption of the single-wall carbon
nanotube to verify intrinsic thermal applications
Light-Sci. Appl. 4 8
[26] Zhang X, Yang F, Zhao D, Cai L, Luan P, Zhang
Q, Zhou W, Zhang N, Fan Q, Wang Y, Liu H,
Zhou W and Xie S 2014 Temperature dependent
Raman spectra of isolated suspended single-
walled carbon nanotubes Nanoscale 6 3949-53
[27] Zhou W B, Fan Q X, Zhang Q, Li K W, Cai L,
Gu X G, Yang F, Zhang N, Xiao Z J, Chen H L,
Xiao S Q, Wang Y C, Liu H P, Zhou W Y and
Xie S S 2016 Ultrahigh-Power-Factor Carbon
Nanotubes and an Ingenious Strategy for
Thermoelectric Performance Evaluation Small
12 3407-+
1
Supplementary Material
High-fidelity Characterization on Anisotropic Thermal Conductivity of Carbon Nanotube Sheets and on Their Effects of Thermal Enhancement of Nanocomposites
Xiao Zhanga (), Wei Tanb, Fiona Smaila, Michael De Volderc, Norman Fleckb, Adam Boiesa()
a Division of Energy, Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK b Division of Mechanics, Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK c Institute for Manufacturing, Department of Engineering, University of Cambridge, Cambridge, CB3 0FS, UK
————————————
Corresponding author, Xiao Zhang email: [email protected] ; Adam Boies: [email protected]
2
Material and Method
A-1. Calculation of Rayleigh number, average Nusselt number and average convective coefficient.
The Rayleigh number (Ra) was calculated with the following expression:
𝑅𝑎 =𝑔𝛽(𝑇−𝑇𝑔)𝑤3
𝜐𝛼 (S1)
here, standard gravity 𝑔 = 9.8 𝑚/𝑠2; β represents the expansion coefficient, for an ideal gas, 𝛽 =1
𝑇; T and 𝑇𝑔 represent
the temperature at sample surface and free gas out of boundary layer; 𝑤 represents the width of sample; 𝜐 and 𝛼 represent
the kinematic viscosity and heat diffusivity of gas.
Based on our experiment results and the thermophysical properties data in Table S1 by temperature interpolation., the Ra
was around 30-400, which is far below the critical Ra=109 to develop a turbulent boundary layer.
Thus, the average Nusselt number (𝑁𝑢̅̅ ̅̅ ) was calculated based on the following correlation[1]:
𝑁𝑢̅̅ ̅̅ =ℎ𝑐̅̅ ̅𝑤
𝜅=
4
3(
𝐺𝑟
4)
1
4𝑔(𝑃𝑟) (S2)
here, ℎ𝑐̅̅ ̅ and 𝜅 represent the average convective coefficient and thermal conductivity of gas, Grashof number 𝐺r =
gβ(T−Tg)w3
υ2 , Prandtl number 𝑃𝑟 =𝜐
𝛼 and 𝑔(𝑃𝑟) is an empirical correlation as follows:
𝑔(𝑃𝑟) =0.75𝑃𝑟1/2
(0.609+1.221𝑃𝑟12+1.238𝑃𝑟)1/4
(S3)
Furthermore, a better fitting result for air could be obtained by using the following empirical correlation[2]:
𝑁𝑢̅̅ ̅̅ = 0.680 +0.670𝑅𝑎1/4
[1+(0.492/𝑃𝑟)9/16]4/9 (S4)
Table S1. Thermophysical Properties of Gases at Atmospheric Pressure
Gas T of Gas
[K]
Density (𝜌)
[kg/m3]
𝑐𝑝
[kJ/kg/K]
Viscosity (𝜇)
[Pa·s]
𝜐
[m2/s]
𝛼
[m2/s]
𝜅
[W/K/m]
Calculated ℎ𝑐̅̅ ̅
[W/K/m2]
Air[1]
250 1.39 1.006 1.60E-05 1.14E-05 1.59E-05 2.23E-02 4.6-8.2 (Eq.S2)
7.1-10.8 (Eq.S4) 300 1.16 1.007 1.85E-05 1.59E-05 2.25E-05 2.63E-02
350 1.00 1.009 2.08E-05 2.09E-05 2.99E-05 3.00E-02
He[1]
260 0.19 5.193 1.80E-05 9.60E-05 1.41E-04 1.37E-01 9.1-16.9 (Eq.S2)
23.3-31.5 (Eq.S4) 300 0.16 5.193 1.99E-05 1.22E-04 1.80E-04 1.52E-01
400 0.12 5.193 2.43E-05 1.99E-04 2.95E-04 1.87E-01
Ar[3]
280 1.72 0.521 2.16E-05 1.26E-05 1.89E-05 1.69E-02
2.9-5.8 (Eq.S2)
4.5-7.5 (Eq.S4)
300 1.60 0.521 2.29E-05 1.43E-05 2.15E-05 1.79E-02
320 1.52 0.521 2.42E-05 1.59E-05 2.39E-05 1.89E-02
340 1.40 0.521 2.54E-05 1.82E-05 2.73E-05 1.99E-02
3
A-2. SEM image of CNT mat and CNT mat reinforced composite.
Figure S-1 SEM image of pure CNT mat for directly measurement and thermal enhancement filler in CNT reinforced polymer
composites.
Figure S-2 SEM image of CNT reinforced polymer composites with 24 vol% of CNTs.
4
Figure S-3 (a) In CNT mat reinforced composite, with epoxy increasing in composite, 𝜅𝐴 normalized by CNT strips’ width (𝑤) and
thickness (𝑡𝐶𝑁𝑇) preserved its value in pure CNT mat; (b) with CNT volume fraction increase in composite, 𝜅𝑐𝑝𝑒𝑓𝑓
kept increasing.
References
[1] Bergman T L 2011 Introduction to Heat Transfer: Wiley
[2] Churchill S W and Chu H H S 1975 CORRELATING EQUATIONS FOR LAMINAR AND TURBULENT
FREE CONVECTION FROM A VERTICAL PLATE Int. J. Heat Mass Transf. 18 1323-9
[3] Haynes W M 2014 CRC Handbook of Chemistry and Physics, 95th Edition: CRC Press